Glowing transparent shader, Eevee and Cycle render different result
The closest I've been able to manage to getting EEVEE and Cycles to appear the same:

I've removed the emission shader but enabled emission directly from the cycle shader and then moved the transparent shader to where you had the emission shader.
This doesn't give exactly the effect you were after but it at least matches cycles and EEVEE as closely as possible.  Perhaps it will give you a starting point for what you did want?
EEVEE:

Cycles (viewport):

For cycles I enabled noise reduction in the viewport render mode: